Topological additive numbering of directed acyclic graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f a labeling of its vertices with positive integers; denote by S(v) the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering if S(u)